Editing Derivative algebra (abstract algebra)

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In [[abstract algebra]], a '''derivative algebra''' is an [[algebraic structure]] of the signature   
    
:<''A'', ·, +, ', 0, 1, <sup>D</sup>>   
    
where    

:<''A'', ·, +, ', 0, 1>   

is a [[Boolean algebra (structure)|Boolean algebra]] and <sup>D</sup> is a [[unary operator]], the '''derivative operator''', satisfying the identities:    
    
# 0<sup>D</sup> = 0    
# ''x''<sup>DD</sup> ≤ ''x'' + ''x''<sup>D</sup>    
# (''x'' + ''y'')<sup>D</sup> = ''x''<sup>D</sup> + ''y''<sup>D</sup>.  

x<sup>D</sup> is called the '''[[derivative]]''' of x. Derivative algebras provide an algebraic abstraction of the '''[[derived set (mathematics)|derived set]]''' operator in [[topological space|topology]]. They also [[Lindenbaum–Tarski algebra|play the same role]] for the [[modal logic]] ''wK4'' = ''K''&nbsp;+ (''p''∧□''p''&nbsp;→&nbsp;□□''p'') that [[Boolean algebra (structure)|Boolean algebra]]s play for ordinary [[propositional logic]].   

==References==   
* Esakia, L., ''Intuitionistic logic and modality via topology'', [[Annals of Pure and Applied Logic]], 127 (2004) 155-170
* [[J.C.C. McKinsey|McKinsey, J.C.C.]] and [[Alfred Tarski|Tarski, A.]], ''The Algebra of Topology'', [[Annals of Mathematics]], 45 (1944) 141-191

[[Category:Algebras]]
[[Category:Boolean algebra]]
[[Category:Topology]]


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